Series 7 – Fitting a Logistic Regression in Bayes

Biostatistics Tutorial R Bayesian Methods JAGS/Stan

How to fit a Logistic Regression using Bayesian Methods
The advantage of using Bayes to overcome the Separation in Logistic Regression

Hai Nguyen
March 20, 2022

Logistic regression

Prediction of gender by hight and weight. Probability close to 0.5 (balanced data)

mydf <- read.csv(file = 'data/HtWtData300.csv')
head(mydf)
  male height weight
1    0   64.0  136.4
2    0   62.3  215.1
3    1   67.9  173.6
4    0   64.2  117.3
5    0   64.8  123.3
6    0   57.5   96.5

Probability close to 0.5

table(mydf$male)

  0   1 
152 148 
prop.table(table(mydf$male))

        0         1 
0.5066667 0.4933333

Firstly, we try to fit with glm

fmod1 <- glm(male ~ height + weight, 
             family = "binomial", data = mydf)
summary(fmod1)

Call:
glm(formula = male ~ height + weight, family = "binomial", data = mydf)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-2.7751  -0.4674  -0.0165   0.3655   3.1906  

Coefficients:
              Estimate Std. Error z value Pr(>|z|)    
(Intercept) -59.994880   7.136907  -8.406  < 2e-16 ***
height        0.890858   0.108873   8.183 2.78e-16 ***
weight        0.005320   0.005225   1.018    0.309    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 415.83  on 299  degrees of freedom
Residual deviance: 189.49  on 297  degrees of freedom
AIC: 195.49

Number of Fisher Scoring iterations: 6

\(\Rightarrow\) We will see the estimates as same as the Bayesian method.

plot(mydf$height, 
     mydf$weight,pch=16, col="blue")
points(mydf$height[mydf$male==1], 
       mydf$weight[mydf$male==1], 
       col="orange", pch=16)
legend("topleft", legend=c("Female","Male"), 
       col=c("blue","orange"), pch=16)

Model for Dichotomous y with multiple metric predictors set up in Stan

But firstly, we look at the simplified modelString of Stan:

# Not eval
modelString <- "
  data{
    int<lower=0> N;
    real xc[N];
    int xb[N];
    int<lower=0,upper=1> y[N]; //Binary outcome
  }
  parameters{
    real b0;
    real b1;
    real b2;
  }
  model{
    b0 ~ normal(0,1);
    b1 ~ normal(0,1);
    b2 ~ normal(0,1);
    y  ~ bernoulli_logit(b0 + b1*xc + b2*xb);
  }
"

Then move on the destination one:

modelString <- "
  data {
    int<lower=1> Ntotal; // num of observations
    int<lower=1> Nx;     // num of predictors
    int<lower=0, upper=1> y[Ntotal];
    matrix[Ntotal, Nx] x;
  }
  
  transformed data {
    vector[Nx] meanX;
    vector[Nx] sdX;
    matrix[Ntotal, Nx] zx; // normalized

    for ( j in 1:Nx ) {
      meanX[j] = mean(x[,j]);
      sdX[j] = sd(x[,j]);
        for (i in 1:Ntotal) {
          zx[i,j] = (x[i,j] - meanX[j]) / sdX[j];
        }
      }
  }
    
  parameters {
    real zbeta0;
    vector[Nx] zbeta;
  }

  transformed parameters{
    vector[Ntotal] mu;
    mu = zbeta0 + zx * zbeta;  // matrix product
  }

  model {
    zbeta0 ~ normal(0, 2);
    zbeta  ~ normal(0, 2);
    y ~ bernoulli_logit(mu);
  }

  generated quantities { 
    // Transform to original scale:
    real beta0; 
    vector[Nx] beta;
    // .* and ./ are element-wise product and division
    beta0 = zbeta0 - sum(zbeta .* meanX ./ sdX);
    beta = zbeta ./ sdX;
  } 
"
stanDsoLogistic <- stan_model(model_code=modelString)
# save(stanDsoLogistic, file = "data/stanLogisticDso.Rdata")
load(file = "data/stanLogisticDso.Rdata")

Fit model

heightWeightDataList <- list(Ntotal = nrow(mydf),
                             y = mydf$male,
                             x = cbind(mydf$height, mydf$weight),
                             Nx = 2)
fit <- sampling(stanDsoLogistic,
                data = heightWeightDataList, 
                pars = c('beta0', 'beta'),
                iter = 5000, chains = 2, cores = 2
)

Analyze fitted model using shinystan

launch_shinystan(fit)

Analyze parameters.

stan_ac(fit, separate_chains = T)

pairs(fit)

plot(fit)

plot(fit,pars=c("beta"))

plot(fit,pars=c("beta[2]"))

summary(fit)$summary[,c(1,4,8)]
                 mean          2.5%        97.5%
beta0   -59.074628416 -7.332501e+01 -46.74736282
beta[1]   0.876102572  6.866473e-01   1.08922356
beta[2]   0.005720759 -4.493517e-03   0.01585729
lp__    -97.682840004 -1.008467e+02 -96.28774967

Parameter \(\beta_2\) is not significant with 95% HDI.

stan_dens(fit)

estimBetas<-summary(fit)$summary[1:3,1]

Plot the data with separating hyperplane.

plot(mydf$height, mydf$weight, pch=16, col="blue")
points(mydf$height[mydf$male==1], mydf$weight[mydf$male==1], col = "orange", pch = 16)
lines(mydf$height, -(estimBetas[1]+estimBetas[2]*mydf$height)/estimBetas[3])
legend("topleft", legend = c("Female","Male"), col = c("blue","orange"), pch=16)

Prediction of gender by hight and weight. Probability close to extreme (imbalanced sample)

Now try to remove almost all males from the sample to see what may happen when there are few 1’s observed.

In the original sample the proportion of males is:

mean(mydf$male)
[1] 0.4933333

Sample with females is

females <- mydf[mydf$male == 0,]

Select first 15 males.

males <- mydf[mydf$male == 1,][1:15,] # just 15 males (originally was ~150)
mydf_sparse <- rbind(males,females)
rownames(mydf_sparse) <- NULL
head(mydf_sparse, 20)
   male height weight
1     1   67.9  173.6
2     1   70.2  191.1
3     1   71.1  193.9
4     1   66.5  127.1
5     1   75.1  204.4
6     1   64.6  143.4
7     1   69.2  124.4
8     1   68.1  140.9
9     1   72.6  164.7
10    1   71.5  193.6
11    1   76.0  180.0
12    1   69.7  155.0
13    1   73.3  188.2
14    1   68.3  178.6
15    1   70.8  207.3
16    0   64.0  136.4
17    0   62.3  215.1
18    0   64.2  117.3
19    0   64.8  123.3
20    0   57.5   96.5

Fit sparse model

fmod2 <- glm(male ~ height + weight, family = "binomial", data = mydf_sparse)
summary(fmod2)

Call:
glm(formula = male ~ height + weight, family = "binomial", data = mydf_sparse)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-1.28353  -0.17117  -0.06829  -0.01696   3.15051  

Coefficients:
             Estimate Std. Error z value Pr(>|z|)    
(Intercept) -84.24302   20.03395  -4.205 2.61e-05 ***
height        1.25377    0.30728   4.080 4.50e-05 ***
weight       -0.01190    0.01248  -0.953     0.34    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 100.909  on 166  degrees of freedom
Residual deviance:  35.582  on 164  degrees of freedom
AIC: 41.582

Number of Fisher Scoring iterations: 8
heightWeightSparseDataList<-list(Ntotal=nrow(mydf_sparse),
                                 y=mydf_sparse$male,
                                 x=cbind(mydf_sparse$height, mydf_sparse$weight),
                                 Nx=2)
fit_sparse <- sampling(stanDsoLogistic,
                       data=heightWeightSparseDataList, 
                       pars=c('beta0', 'beta'),
                       iter=5000, chains = 2, cores = 2
)
stan_ac(fit_sparse, separate_chains = T)

pairs(fit_sparse)

plot(fit_sparse)

plot(fit_sparse,pars=c("beta"))

plot(fit_sparse,pars=c("beta[2]"))

Compare summary of the two studies

rbind(beta0reg=summary(fit)$summary[1,c(1,3)],
      beta0glm=c(summary(fmod1)$coefficients[1,1],summary(fmod1)$coefficients[1,2]*sqrt(dim(mydf)[1])),
      beta0sparce=summary(fit_sparse)$summary[1,c(1,3)],
      beta0sparceglm=c(summary(fmod2)$coefficients[1,1],summary(fmod2)$coefficients[1,2]*sqrt(dim(mydf_sparse)[1])))
                    mean         sd
beta0reg       -59.07463   6.749991
beta0glm       -59.99488 123.614858
beta0sparce    -65.48010  12.250007
beta0sparceglm -84.24302 258.895713
rbind(beta1reg=summary(fit)$summary[2,c(1,3)],
      beta1glm=c(summary(fmod1)$coefficients[2,1],summary(fmod1)$coefficients[2,2]*sqrt(dim(mydf)[1])),
      beta1sparce=summary(fit_sparse)$summary[2,c(1,3)],
      beta1sparceglm=c(summary(fmod2)$coefficients[2,1],summary(fmod2)$coefficients[2,2]*sqrt(dim(mydf_sparse)[1])))
                    mean        sd
beta1reg       0.8761026 0.1030361
beta1glm       0.8908578 1.8857272
beta1sparce    0.9664459 0.1882613
beta1sparceglm 1.2537728 3.9708903
rbind(beta2reg=summary(fit)$summary[3,c(1,3)],
      beta2glm=c(summary(fmod1)$coefficients[3,1],summary(fmod1)$coefficients[3,2]*sqrt(dim(mydf)[1])),
      beta2sparce=summary(fit_sparse)$summary[3,c(1,3)],
      beta2sparceglm=c(summary(fmod2)$coefficients[3,1],summary(fmod2)$coefficients[3,2]*sqrt(dim(mydf_sparse)[1])))
                       mean          sd
beta2reg        0.005720759 0.005286843
beta2glm        0.005319759 0.090499569
beta2sparce    -0.007041950 0.010449550
beta2sparceglm -0.011900490 0.161304086
rbind(beta0reg=summary(fit)$summary[1,c(4,8)],
      beta0sparce=summary(fit_sparse)$summary[1,c(4,8)])
                 2.5%     97.5%
beta0reg    -73.32501 -46.74736
beta0sparce -90.86386 -43.48094
rbind(beta1reg=summary(fit)$summary[2,c(4,8)],
      beta1sparce=summary(fit_sparse)$summary[2,c(4,8)])
                 2.5%    97.5%
beta1reg    0.6866473 1.089224
beta1sparce 0.6268317 1.359368
rbind(beta2reg=summary(fit)$summary[3,c(4,8)],
      beta2sparce=summary(fit_sparse)$summary[3,c(4,8)])
                    2.5%      97.5%
beta2reg    -0.004493517 0.01585729
beta2sparce -0.028395094 0.01216875

HDI of both slopes widened significantly in the sample with more extreme disproportion.
Standard deviations of betas also increase dramatically, especially fit with glm.

Robust logistic regression

Prediction of gender by height and weight. Robust model

Observe the data of the previous section.
Plot male (1) and female (0) groups with respect to weight.

plot(mydf$weight,mydf$male)

In the lower right corner there are some outliers representing heavy females.
Such observations cause bias of model parameters.

plot(mydf$height,mydf$male)

On the plot relative to height outliers are short males.

What does a model robust to such outliers look like?

Robust logistic regression is a mix of “guessing model” and logistic model

\[\mu = \alpha \frac{1}{2} + (1 - \alpha) \ \text{logistic}\Big(\beta_0 + \sum_j \beta_j x_j\Big)\]

Typically \(\alpha\) is small (few outliers showing that predictor points to wrong class).
Select beta distribution as a prior to \(\alpha\) with high concentration near zero: dbeta(1,9).

Argument <- seq(from=0,to=1,by=.01)
plot(Argument, dbeta(Argument, 1, 9), type="l")

The modified model is on the diagram.

modelString= 
"data {                   // ROBUST LOGISTIC REGRESSION
    int<lower=1> Ntotal;  // num of observations
    int<lower=1> Nx;      // num of predictors
    int<lower=0, upper=1> y[Ntotal];
    matrix[Ntotal, Nx] x;
}
transformed data {
    vector[Nx] meanX;
    vector[Nx] sdX;
    matrix[Ntotal, Nx] zx;  // normalized
    
    for ( j in 1:Nx ) {
        meanX[j] = mean(x[,j]);
        sdX[j] = sd(x[,j]);
        for ( i in 1:Ntotal ) {
            zx[i,j] = ( x[i,j] - meanX[j] ) / sdX[j];
        }
    }
}
parameters {
    real zbeta0;
    vector[Nx] zbeta;
    real<lower=0,upper=1> guess;  // mixture param
}
transformed parameters{
    vector[Ntotal] mu;
    for ( i in 1:Ntotal ) {
        mu[i] = guess * (1/2.0) + (1-guess) * inv_logit(zbeta0 + zx[i,] * zbeta);
    }
}
model {
    zbeta0 ~ normal(0, 2);
    zbeta  ~ normal(0, 2);
    guess ~ beta(1, 9);
    y ~ bernoulli(mu);
}
generated quantities { 
    // Transform to original scale:
    real beta0; 
    vector[Nx] beta;
    // .* and ./ are element-wise product and division
    beta0 =  zbeta0 - sum( zbeta .* meanX ./ sdX );
    beta =  zbeta ./ sdX;
}
"
stanDsoRobustLogistic <- stan_model(model_code=modelString)
# save(stanDsoRobustLogistic, file = "data/stanRobustLogisticDso.Rds")
load("data/stanRobustLogisticDso.Rds")

Run robust MCMC with the hight/weight data.

fitRobust <- sampling(stanDsoRobustLogistic,
                data=heightWeightDataList, 
                pars=c('beta0', 'beta', 'guess'),
                iter=5000, chains = 2, cores = 2)

Analyze results.

stan_ac(fitRobust, separate_chains = T)

pairs(fitRobust)

plot(fitRobust)

plot(fitRobust,pars=c("beta[1]"))

plot(fitRobust,pars=c("beta[2]"))

plot(fitRobust,pars=c("guess"))

rbind(summary(fitRobust)$summary[,c(1,4,7)],
      summary(fit)$summary[,c(1,4,8)]
)
                 mean          2.5%           75%
beta0   -6.759671e+01 -9.033603e+01  -60.44830982
beta[1]  9.987992e-01  7.405794e-01    1.08941932
beta[2]  7.863251e-03 -4.898361e-03    0.01200465
guess    3.520801e-02  1.562728e-03    0.04978482
lp__    -1.020473e+02 -1.058665e+02 -100.91970991
beta0   -5.907463e+01 -7.332501e+01  -46.74736282
beta[1]  8.761026e-01  6.866473e-01    1.08922356
beta[2]  5.720759e-03 -4.493517e-03    0.01585729
lp__    -9.768284e+01 -1.008467e+02  -96.28774967

Note positive correlation between guess and the slope \(\beta_1\). The more outliers the higher is the slope.

Since \(\beta_2\) does not seem significant. Now, we fit both models with height as only predictor.

heightWeightDataList<-list(Ntotal=nrow(mydf),
                          y=mydf$male,
                          x=cbind(mydf$height),
                          Nx=1)


fit <- sampling(stanDsoLogistic,
                data=heightWeightDataList, 
                pars=c('beta0', 'beta'),
                iter=5000, chains = 2, cores = 2
)
fitRobust <- sampling(stanDsoRobustLogistic,
                data=heightWeightDataList, 
                pars=c('beta0', 'beta', 'guess'),
                iter=5000, chains = 2, cores = 2
)
pairs(fit)

pairs(fitRobust)

plot(fit)

plot(fit,pars=c("beta"))

plot(fitRobust)

plot(fitRobust,pars=c("beta[1]","guess"))

plot(fitRobust,pars=c("guess"))

rbind(summary(fitRobust)$summary[,c(1,4,7)],
      summary(fit)$summary[,c(1,4,8)]
)
                 mean          2.5%           75%
beta0    -66.86054035 -8.942821e+01  -59.89807734
beta[1]    1.00658061  7.500421e-01    1.09452331
guess      0.03199073  1.072547e-03    0.04549365
lp__    -102.27095400 -1.058139e+02 -101.30279477
beta0    -59.24314304 -7.275107e+01  -47.12218747
beta[1]    0.89230096  7.081439e-01    1.09712292
lp__     -97.71023494 -1.002870e+02  -96.75923890

Compare probabilities predicted by logistic regression and robust logistic regression.

# Coefficients
meanBeta0Robust<-summary(fitRobust)$summary[1,1]
meanBeta1Robust<-summary(fitRobust)$summary[2,1]
guess<-summary(fitRobust)$summary[3,1]
meanBeta0<-summary(fit)$summary[1,1]
meanBeta1<-summary(fit)$summary[2,1]

#Linear predictors and probabilities
linPredRobust_Male.Height<-meanBeta0Robust+meanBeta1Robust*mydf$height
pRobustMail_height<-guess/2+(1-guess)*exp(linPredRobust_Male.Height)/(1+exp(linPredRobust_Male.Height))
linPred_Male.Height<-meanBeta0+meanBeta1*mydf$height
pMail_height<-exp(linPred_Male.Height)/(1+exp(linPred_Male.Height))

# Plot
plot(mydf$height,mydf$male,pch=16)
points(mydf$height,pRobustMail_height,col="orange",pch=16)
points(mydf$height,pMail_height,col="cyan",pch=16)
legend("topleft",
       legend=c("Actual","Prob Logistic","Prob. Robust"),
       col=c("black","cyan","orange"), pch=16)

Logistic probability is more extreme: higher in the area of low-height male observations because it is biased by an outlier of short male below the average female height. But the far tails are closer to 0.
Robust logistic regression did not react to this observation as much!!
Robust model does not produce probabilities too close to 0 and 1.

Prediction of gender by height and weight with sparse data. Robust model

Repeat the same comparison with the sparse data.

Create data list with one predictor.

heightWeightSparseDataList <- list(Ntotal = nrow(mydf_sparse),
                                   y = mydf_sparse$male,
                                   x = cbind(mydf_sparse$height),
                                   Nx = 1)

Fit both models with only one predictor height

fitSparse <- sampling(stanDsoLogistic,
                       data=heightWeightSparseDataList, 
                       pars=c('beta0', 'beta'),
                       iter=5000, chains = 2, cores = 2)

fitSparseRobust <- sampling(stanDsoRobustLogistic,
                data=heightWeightSparseDataList, 
                pars=c('beta0', 'beta', 'guess'),
                iter=5000, chains = 2, cores = 2)

Analyze the models.

pairs(fitSparse)

pairs(fitSparseRobust)

plot(fitSparse)

plot(fitSparse,pars=c("beta"))

plot(fitSparseRobust)

plot(fitSparseRobust,pars=c("beta[1]","guess"))

plot(fitSparseRobust,pars=c("guess"))

rbind(summary(fitSparseRobust)$summary[,c(1,4,7)],
      summary(fitSparse)$summary[,c(1,4,8)]
)
                mean          2.5%          75%
beta0   -65.43711171 -9.393761e+01 -56.03873197
beta[1]   0.94912626  6.055210e-01   1.07111437
guess     0.01689194  5.152762e-04   0.02448201
lp__    -28.85479455 -3.232575e+01 -27.89492051
beta0   -63.02455548 -8.871818e+01 -42.69724337
beta[1]   0.91384987  6.114798e-01   1.29416346
lp__    -23.14591098 -2.575415e+01 -22.20307431

Make plot of probabilities.

# Coefficients
meanBeta0Robust<-summary(fitSparseRobust)$summary[1,1]
meanBeta1Robust<-summary(fitSparseRobust)$summary[2,1]
guess<-summary(fitSparseRobust)$summary[3,1]
meanBeta0<-summary(fitSparse)$summary[1,1]
meanBeta1<-summary(fitSparse)$summary[2,1]

#Linear predictors and probabilities
linPredRobust_Male.Height<-meanBeta0Robust+meanBeta1Robust*mydf_sparse$height
pRobustMail_height<-guess/2+(1-guess)*exp(linPredRobust_Male.Height)/(1+exp(linPredRobust_Male.Height))
linPred_Male.Height<-meanBeta0+meanBeta1*mydf_sparse$height
pMail_height<-exp(linPred_Male.Height)/(1+exp(linPred_Male.Height))

# Plot
plot(mydf_sparse$height,mydf_sparse$male,pch=16)
points(mydf_sparse$height,pRobustMail_height,col="orange",pch=16)
points(mydf_sparse$height,pMail_height,col="cyan",pch=16)
legend("topleft",
       legend=c("Actual","Prob Logistic","Prob. Robust"),
       col=c("black","cyan","orange"),pch=16)

Anesthesia example

This data example is from library DAAG.
Thirty patients were given an anesthetic agent maintained at a predetermined concentration level (conc) for 15 minutes before making an incision. It was then noted whether the patient moved (1), i.e. jerked or twisted.

Data

library(DAAG)
head(anesthetic)
  move conc   logconc nomove
1    0  1.0 0.0000000      1
2    1  1.2 0.1823216      0
3    0  1.4 0.3364722      1
4    1  1.4 0.3364722      0
5    1  1.2 0.1823216      0
6    0  2.5 0.9162907      1

Use column move as response and column logconc as predictor.

Prepare the data.

dataListAnesthetic <- list(Ntotal=nrow(anesthetic),
                           y=anesthetic$move,
                           x=cbind(anesthetic$logconc),
                           Nx=1)

Logistic model by glm()

logRegr <- glm(move ~ logconc, 
               data=anesthetic,
               family="binomial")
summary(logRegr)

Call:
glm(formula = move ~ logconc, family = "binomial", data = anesthetic)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-2.1477  -0.6962  -0.1209   0.7586   1.7528  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)   
(Intercept)   0.8077     0.5709   1.415  0.15715   
logconc      -6.2457     2.2618  -2.761  0.00575 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 41.455  on 29  degrees of freedom
Residual deviance: 28.007  on 28  degrees of freedom
AIC: 32.007

Number of Fisher Scoring iterations: 5
predLogRegr <- predict(logRegr, type="response")

Running chains

Run MCMC using logistic and robust logistic models.

fitAnesth <- sampling(stanDsoLogistic,
                data=dataListAnesthetic, 
                pars=c('beta0', 'beta'),
                iter=5000, chains = 2, cores = 2
)

fitRobustAnesth <- sampling(stanDsoRobustLogistic,
                             data=dataListAnesthetic, 
                              pars=c('beta0', 'beta', 'guess'),
                              iter=5000, chains = 2, cores = 2)

Analysis of logistic model

Extract diagnostics

summary(fitAnesth)$summary[,c(1,4,8:10)]
               mean        2.5%      97.5%    n_eff      Rhat
beta0     0.8198095  -0.2232116   2.005003 3877.299 0.9998678
beta[1]  -6.1749593 -10.8825706  -2.377365 3117.371 1.0000402
lp__    -15.4911255 -18.3350197 -14.471061 1721.835 1.0022634
stan_ac(fitAnesth, separate_chains = T)

stan_trace(fitAnesth)

pairs(fitAnesth,pars=c("beta0","beta"))

plot(fitAnesth)

Analysis of robust logistic model

summary(fitRobustAnesth)$summary[,c(1,4,8:10)]
                mean          2.5%       97.5%    n_eff      Rhat
beta0     1.02591748  -0.206024406   2.6568493 2882.397 1.0010848
beta[1]  -7.40456341 -14.191363295  -2.7011459 2542.829 1.0008553
guess     0.09682537   0.003829574   0.2914329 4230.349 0.9996954
lp__    -19.49792255 -22.837487799 -18.0184837 1903.465 1.0013869
stan_ac(fitRobustAnesth, separate_chains = T)

stan_trace(fitRobustAnesth)

pairs(fitRobustAnesth,pars=c("beta0","beta","guess"))

plot(fitRobustAnesth)

plot(fitRobustAnesth,pars=c("guess"))

Parameter guess is almost \(10%\). This means that there should be a significant difference between the two models.
However, 95% HDI almost covers \(0\).

Comparison of logistic and robust logistic models

Compare intercepts.

rbind(Logistic=summary(fitAnesth)$summary[1,c(1,4,8)],
      Robust=summary(fitRobustAnesth)$summary[1,c(1,4,8)])
              mean       2.5%    97.5%
Logistic 0.8198095 -0.2232116 2.005003
Robust   1.0259175 -0.2060244 2.656849

Compare slopes.

rbind(Logistic=summary(fitAnesth)$summary[2,c(1,4,8)],
      Robust=summary(fitRobustAnesth)$summary[2,c(1,4,8)])
              mean      2.5%     97.5%
Logistic -6.174959 -10.88257 -2.377365
Robust   -7.404563 -14.19136 -2.701146

Compare probabilities.

# Coefficients
meanBeta0Robust<-summary(fitRobustAnesth)$summary[1,1]
meanBeta1Robust<-summary(fitRobustAnesth)$summary[2,1]
guess<-summary(fitRobustAnesth)$summary[3,1]
meanBeta0<-summary(fitAnesth)$summary[1,1]
meanBeta1<-summary(fitAnesth)$summary[2,1]

#Linear predictors and probabilities
linPredRobust_Move<-meanBeta0Robust+meanBeta1Robust*anesthetic$logconc
pRobustMove<-guess/2+(1-guess)*exp(linPredRobust_Move)/(1+exp(linPredRobust_Move))
linPred_Move<-meanBeta0+meanBeta1*anesthetic$logconc
pMove<-exp(linPred_Move)/(1+exp(linPred_Move))

# Plot
plot(anesthetic$logconc,anesthetic$move,pch=16)
points(anesthetic$logconc,pRobustMove,col="orange",pch=15)
points(anesthetic$logconc,pMove,col="cyan",pch=17)
points(anesthetic$logconc,predLogRegr,col="purple",pch=25)
legend("topright",
       legend=c("Actual","Prob Logistic","Prob. Robust","Glm"),
       col=c("black","cyan","orange","purple"),pch=c(16,17,15,25))

Again, robust method does not return extreme probabilities.

Softmax regression

Softmax (or multinomial logistic regression) is a generalization of logistic regression to the case where we want to handle multiple classes. In logistic regression, the response was binary: \(y^{(i)} \in \{0,1\}\), meanwhile softmax handles \(y^{(i)} \in \{1, ..., K\}\)

It is based on modification of odds ratio from
\[\frac{p}{1-p}\] to \[\frac{p_s}{p_r},\]
where \(p_r\) is probability of one of several classes selected as reference (for example, control group).

The model changes very little from logistic regression and is shown on this diagram.

Because adding constants to coefficients does not change softmax the reference class coefficients are assumed equal to zero.

Simulated data from the book

The data SoftmaxRegData1.csv are from Kruschke, 2015 chapter 22.

myData <- read.csv(file="data/SoftmaxRegData2.csv" )
head(myData)
           X1         X2 Y
1 -0.08714736 -1.0813422 2
2 -0.72256565 -1.5838631 2
3  0.17918961  0.9717904 3
4 -1.15975176  0.5026244 3
5 -0.72711762  1.3757045 3
6  0.53341559  1.7746506 3
table(myData$Y, useNA = "always")

   1    2    3    4 <NA> 
  58  145  139  133    0 
idx2<-myData$Y==2
idx3<-myData$Y==3
idx4<-myData$Y==4

plot(myData$X1,myData$X2,pch=16)
points(myData$X1[idx2],myData$X2[idx2],pch=16,col="orange")
points(myData$X1[idx3],myData$X2[idx3],pch=16,col="cyan")
points(myData$X1[idx4],myData$X2[idx4],pch=16,col="magenta")

Prepare data list

dataListSoftmax<-list(N=nrow(myData),  # num of observations
                          K=max(myData$Y), # num of groups
                          y=myData$Y,
                          x=cbind(x1 = myData$X1, x2 = myData$X2),
                          D=2)  # num of predictiors

Softmax model

Describe the model.

modelString="
data {
    int<lower=2> K;  // num of groups
    int<lower=0> N;  // num of observations
    int<lower=1> D;  // num of predictors 
    int<lower=1,upper=K> y[N];
    matrix[N, D] x;
}
transformed data {
    row_vector[D] zeros;
    row_vector[D] x_m;  // x means
    row_vector[D] x_sd; // x standard deviations
    matrix[N, D] zx;    // normalized x
    zeros = rep_row_vector(0, D); // coefficients are zeros for the baseline class
    for (j in 1:D) {
        x_m[j] = mean(x[,j]);
        x_sd[j] = sd(x[,j]);
        zx[,j] = (x[,j] - x_m[j]) / x_sd[j];
    }
}
parameters {
    matrix[K-1,D] zbeta_raw;  // K-1 makes model identifiable
    vector[K-1] zbeta0_raw;
}
transformed parameters {
    vector[K] zbeta0;   // intersection coeffs
    matrix[K, D] zbeta; // predictor coeffs
    zbeta0 = append_row(0, zbeta0_raw);
    zbeta = append_row(zeros, zbeta_raw); // add zeros for coefficients of the baseclass
}
model {
    zbeta0_raw ~ normal(0, 5);
    for (k in 1:(K-1))
        zbeta_raw[k,] ~ normal(0, 5);
    for (n in 1:N)
        y[n] ~ categorical(softmax(zbeta0 + zbeta * to_vector(zx[n,]) ));
}
generated quantities {
    vector[K] beta0;
    matrix[K, D] beta;
    // transform zbetas to original betas:
    for (k in 1:K) {
        beta0[k] = zbeta0[k];
        for (j in 1:D) {
            beta0[k] = beta0[k] - zbeta[k,j] * x_m[j] / x_sd[j];
            beta[k,j] = zbeta[k,j] / x_sd[j];
        }
     }
}
"

Create DSO.

modelSoftmax <- stan_model(model_code=modelString)
save(modelSoftmax, file = "data/stanSoftmaxDso.Rds")
load("data/stanSoftmaxDso.Rds")
fit <- sampling(modelSoftmax,
                data=dataListSoftmax,
                pars=c('beta0', 'beta'),
                iter=5000, chains = 2, cores = 2)

Analysis

Analyze fitted model using shinystan, but check directly for purposely demonstration.

summary(fit)$summary[,c(1,4,8:10)]
                  mean        2.5%        97.5%    n_eff      Rhat
beta0[1]     0.0000000    0.000000    0.0000000      NaN       NaN
beta0[2]    -3.8747761   -5.287726   -2.6498584 2424.806 1.0004543
beta0[3]    -3.4383430   -4.758058   -2.2832007 3046.880 1.0003798
beta0[4]    -3.8239785   -5.174666   -2.6048550 2777.099 1.0014128
beta[1,1]    0.0000000    0.000000    0.0000000      NaN       NaN
beta[1,2]    0.0000000    0.000000    0.0000000      NaN       NaN
beta[2,1]   -5.5258762   -7.266858   -4.0493728 2509.455 1.0003697
beta[2,2]   -5.3813396   -7.078252   -3.9099912 2562.798 1.0001610
beta[3,1]   -1.4477120   -2.502389   -0.4137045 3499.414 1.0012289
beta[3,2]    8.7504970    6.700614   11.0491097 2771.076 0.9998857
beta[4,1]    8.1935589    6.190566   10.3753359 3002.059 1.0013696
beta[4,2]   -0.7384386   -1.710137    0.1956379 3498.552 1.0011204
lp__      -123.5101370 -128.879940 -120.2345545 2018.045 1.0005849

The model has 4 classes and 2 predictors. The returned coefficients form a matrix \[\lambda_{i,1} = \beta_{0,1} + \beta{1,1} x_{i,1} + \beta_{2,1} x_{i,2} \\ \lambda_{i,2} = \beta_{0,2} + \beta{1,2} x_{i,1} + \beta_{2,2} x_{i,2} \\ \lambda_{i,3} = \beta_{0,3} + \beta{1,3} x_{i,1} + \beta_{2,31} x_{i,2} \\ \lambda_{i,4} = \beta_{0,4} + \beta{1,4} x_{i,1} + \beta_{2,4} x_{i,2}\]

In this system first class is selected as reference, so \(\beta_{0,1} = \beta_{1,1} = \beta_{2,1} = 0\).

stan_ac(fit, separate_chains = T)

stan_trace(fit)

pairs(fit,pars=c("beta0"))

pairs(fit,pars=c("beta"))

plot(fit)

Classification

To predict classes use formula for probability of class \(k\) \[p_k = \frac{e^{\lambda_k}}{\sum_s e^{\lambda_s}}.\]

Create matrix of coefficients.

SoftmaxCoeff<-summary(fit)$summary[1:12,c(1)]
SoftmaxCoeff<-cbind(SoftmaxCoeff[1:4],matrix(SoftmaxCoeff[-(1:4)],ncol=2,byrow=T))
rownames(SoftmaxCoeff)<-paste0("Class",1:4)
SoftmaxCoeff
            [,1]      [,2]       [,3]
Class1  0.000000  0.000000  0.0000000
Class2 -3.874776 -5.525876 -5.3813396
Class3 -3.438343 -1.447712  8.7504970
Class4 -3.823979  8.193559 -0.7384386

Create linear predictors.

head(myData)
           X1         X2 Y
1 -0.08714736 -1.0813422 2
2 -0.72256565 -1.5838631 2
3  0.17918961  0.9717904 3
4 -1.15975176  0.5026244 3
5 -0.72711762  1.3757045 3
6  0.53341559  1.7746506 3
linPredictors<-apply(SoftmaxCoeff[,-1],1,function(z) z%*%t(myData[,1:2]))
dim(linPredictors)
[1] 475   4
head(linPredictors)
     Class1     Class2     Class3      Class4
[1,]      0   6.300635  -9.336117  0.08445775
[2,]      0  12.516113 -12.813522 -4.75079860
[3,]      0  -6.219714   8.244234  0.75059308
[4,]      0   3.703852   6.077200 -9.87365165
[5,]      0  -3.385171  13.090755 -6.97355432
[6,]      0 -12.497586  14.756843  3.06010158
linPredictors<-t(apply(linPredictors,1,function(z) z+SoftmaxCoeff[,1]))
dim(linPredictors)
[1] 475   4
head(linPredictors)
     Class1      Class2     Class3      Class4
[1,]      0   2.4258589 -12.774460  -3.7395208
[2,]      0   8.6413373 -16.251865  -8.5747771
[3,]      0 -10.0944901   4.805891  -3.0733854
[4,]      0  -0.1709239   2.638857 -13.6976302
[5,]      0  -7.2599471   9.652412 -10.7975328
[6,]      0 -16.3723623  11.318500  -0.7638769

Check calculation for the first row of the data and second class.

row1<-myData[1,]
Class2<-SoftmaxCoeff[2,1]+SoftmaxCoeff[2,2]*row1[1]+SoftmaxCoeff[2,3]*row1[2]
c(Class2,linPredictors[1,2])
$X1
[1] 2.425859

$Class2
[1] 2.425859

Create probabilities

softmaxProb<-exp(linPredictors)/apply(exp(linPredictors),1,sum)
apply(head(softmaxProb),1,sum)
[1] 1 1 1 1 1 1

Predict classes.

predClass<-apply(softmaxProb,1,which.max)
head(predClass)
[1] 2 2 3 3 3 3

Plot predicted classes and compare them with the data.

idx2Pred<-predClass==2
idx3Pred<-predClass==3
idx4Pred<-predClass==4

par(mfrow=c(1,2))
plot(myData$X1,myData$X2,pch=16)
points(myData$X1[idx2],myData$X2[idx2],pch=16,col="orange")
points(myData$X1[idx3],myData$X2[idx3],pch=16,col="cyan")
points(myData$X1[idx4],myData$X2[idx4],pch=16,col="magenta")

plot(myData$X1,myData$X2,pch=16)
points(myData$X1[idx2Pred],myData$X2[idx2Pred],pch=16,col="orange")
points(myData$X1[idx3Pred],myData$X2[idx3Pred],pch=16,col="cyan")
points(myData$X1[idx4Pred],myData$X2[idx4Pred],pch=16,col="magenta")

par(mfrow=c(1,1))

See how different classes are separated by hyperplanes.

Add hyperplane between class 1 and class 2:

plot(myData$X1,myData$X2,pch=16)
points(myData$X1[idx2Pred],myData$X2[idx2Pred],pch=16,col="orange")
points(myData$X1[idx3Pred],myData$X2[idx3Pred],pch=16,col="cyan")
points(myData$X1[idx4Pred],myData$X2[idx4Pred],pch=16,col="magenta")
lines(myData$X1,-(SoftmaxCoeff[2,1]+SoftmaxCoeff[2,2]*myData$X1)/SoftmaxCoeff[2,3],col="grey")

Add hyperplane between class 1 and class 3.

plot(myData$X1,myData$X2,pch=16)
points(myData$X1[idx2Pred],myData$X2[idx2Pred],pch=16,col="orange")
points(myData$X1[idx3Pred],myData$X2[idx3Pred],pch=16,col="cyan")
points(myData$X1[idx4Pred],myData$X2[idx4Pred],pch=16,col="magenta")
lines(myData$X1,-(SoftmaxCoeff[2,1]+SoftmaxCoeff[2,2]*myData$X1)/SoftmaxCoeff[2,3],col="grey")
lines(myData$X1,-(SoftmaxCoeff[3,1]+SoftmaxCoeff[3,2]*myData$X1)/SoftmaxCoeff[3,3],col="grey")

Add hyperplane between class 1 and class 4.

plot(myData$X1,myData$X2,pch=16)
points(myData$X1[idx2Pred],myData$X2[idx2Pred],pch=16,col="orange")
points(myData$X1[idx3Pred],myData$X2[idx3Pred],pch=16,col="cyan")
points(myData$X1[idx4Pred],myData$X2[idx4Pred],pch=16,col="magenta")
lines(myData$X1,-(SoftmaxCoeff[2,1]+SoftmaxCoeff[2,2]*myData$X1)/SoftmaxCoeff[2,3],col="grey")
lines(myData$X1,-(SoftmaxCoeff[3,1]+SoftmaxCoeff[3,2]*myData$X1)/SoftmaxCoeff[3,3],col="grey")
lines(myData$X1,-(SoftmaxCoeff[4,1]+SoftmaxCoeff[4,2]*myData$X1)/SoftmaxCoeff[4,3],col="grey")

Further reading

Corrections

If you see mistakes or want to suggest changes, please create an issue on the source repository.

Reuse

Text and figures are licensed under Creative Commons Attribution CC BY 4.0. Source code is available at https://github.com/hai-mn/hai-mn.github.io, unless otherwise noted. The figures that have been reused from other sources don't fall under this license and can be recognized by a note in their caption: "Figure from ...".

Citation

For attribution, please cite this work as

Nguyen (2022, March 20). HaiBiostat: Series 7 -- Fitting a Logistic Regression in Bayes. Retrieved from https://hai-mn.github.io/posts/2022-03-20-Bayesian methods - Series 7 of 10/

BibTeX citation

@misc{nguyen2022series,
  author = {Nguyen, Hai},
  title = {HaiBiostat: Series 7 -- Fitting a Logistic Regression in Bayes},
  url = {https://hai-mn.github.io/posts/2022-03-20-Bayesian methods - Series 7 of 10/},
  year = {2022}
}